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Remarks on $p$-primary torsion of the Brauer group

Yuan Yang

TL;DR

The paper provides a structural description of the $p$-primary torsion in the Brauer group of smooth proper varieties over an algebraically closed field of characteristic $p>0$, expressing ${\rm Br}(X)[p^\infty]$ as a sum of copies of $\mathbb{Q}_p/\mathbb{Z}_p$ plus a finite-exponent piece controlled by a finite group $J$ and a connected unipotent group $U$. It proves that $U$ vanishes for ordinary varieties, identifies $J$ as the Pontryagin dual of $NS(X)[p^\infty]$ for surfaces, and shows $J=0$ for abelian varieties, with Crew’s formula giving the unipotent dimension in key cases (surfaces and abelian 3-folds). The work uses de Rham–Witt theory, the slope spectral sequence, and the Cart ier–Dieudonné–Raynaud framework to relate $p$-torsion in the Brauer group to slope data $m^{ij}$ and Hodge–Witt numbers $h_W^{ij}$, and it establishes a stability result for unipotent parts under products with ordinary varieties. It also provides criteria ensuring injectivity of the flat-to-crystalline comparison in degree two, under degeneracy and torsion-freeness hypotheses, with applications to abelian varieties and K3 surfaces.

Abstract

For a smooth and proper variety $X$ over an algebraically closed field $k$ of characteristic $p>0$, the group $Br(X)[p^\infty]$ is a direct sum of finitely many copies of $\mathbb{Q}_p/\mathbb{Z}_p$ and an abelian group of finite exponent. The latter is an extension of a finite group $J$ by the group of $k$-points of a connected commutative unipotent algebraic group $U$. In this paper we show that (1) if $X$ is ordinary, then $U = 0$; (2) if $X$ is a surface, then $J$ is the Pontryagin dual of $NS(X)[p^\infty]$; (3) if $X$ is an abelian variety, then $J = 0$. Using Crew's formula, we compute the dimension of $U$ for surfaces and abelian $3$-folds. We show that, if $X$ is ordinary, then the unipotent subgroup of $Br(X\times Y)$ is isomorphic to the unipotent subgroup of $Br(Y)$. Generalizing a result of Ogus, we give a criterion for the injectivity of the canonical map from flat to crystalline cohomology in degree $2$.

Remarks on $p$-primary torsion of the Brauer group

TL;DR

The paper provides a structural description of the -primary torsion in the Brauer group of smooth proper varieties over an algebraically closed field of characteristic , expressing as a sum of copies of plus a finite-exponent piece controlled by a finite group and a connected unipotent group . It proves that vanishes for ordinary varieties, identifies as the Pontryagin dual of for surfaces, and shows for abelian varieties, with Crew’s formula giving the unipotent dimension in key cases (surfaces and abelian 3-folds). The work uses de Rham–Witt theory, the slope spectral sequence, and the Cart ier–Dieudonné–Raynaud framework to relate -torsion in the Brauer group to slope data and Hodge–Witt numbers , and it establishes a stability result for unipotent parts under products with ordinary varieties. It also provides criteria ensuring injectivity of the flat-to-crystalline comparison in degree two, under degeneracy and torsion-freeness hypotheses, with applications to abelian varieties and K3 surfaces.

Abstract

For a smooth and proper variety over an algebraically closed field of characteristic , the group is a direct sum of finitely many copies of and an abelian group of finite exponent. The latter is an extension of a finite group by the group of -points of a connected commutative unipotent algebraic group . In this paper we show that (1) if is ordinary, then ; (2) if is a surface, then is the Pontryagin dual of ; (3) if is an abelian variety, then . Using Crew's formula, we compute the dimension of for surfaces and abelian -folds. We show that, if is ordinary, then the unipotent subgroup of is isomorphic to the unipotent subgroup of . Generalizing a result of Ogus, we give a criterion for the injectivity of the canonical map from flat to crystalline cohomology in degree .

Paper Structure

This paper contains 15 sections, 45 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Review of the de Rham--Witt theory
  4. De Rham--Witt complex and logarithmic cohomology
  5. Coherent $R$-modules
  6. Crew's formula and Ekedahl's inequality
  7. Cartier module of the higher direct image
  8. Applications
  9. Ordinary varieties
  10. Surfaces
  11. Abelian varieties
  12. Product with an ordinary variety
  13. The map from flat to crystalline cohomology
  14. Two long exact sequences
  15. A criterion of injectivity

Key Result

Theorem 1.1

Let $X$ be an ordinary smooth proper variety over $k$. Then the following statements hold: (1) there is an isomorphism of finitely generated torsion $W$-modules (2) there is an isomorphism $\mathrm{Br}(X)[p^\infty]\cong (\mathbb{Q}_p/\mathbb{Z}_p)^{r-\rho}\oplus J$, where $J$ is a finite $p$-group such that $J\otimes W \cong \mathrm{H}^2(X,W\Omega^1_{X/k})[p^{\infty}]$.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1: Artin, Bragg, Olsson
  • Theorem 2.2: Artin, Milne, Berthelot Be
  • Theorem 2.3: Raynaud, Bragg
  • ...and 41 more